- Gila Sher ‘Logical consequence: An Epistemic Outlook’
Introはこうです。Logicのdemarcation problemとかlogical constants絡みで入手。What is logic?というThemeのものはよく集めてた関係から。
The 1st-order thesis, namely, the thesis that logical consequence is standard 1st-order logical consequence, has been widely challenged in recent decades. My own challenge to this thesis in The Bounds of Logic (and related articles) was motivated by what I perceived to be its inadequate philosophical grounding. The bounds of logic are, in an important sense, the bounds of logical constants, yet the bounds of the standard logical constants are specified by enumeration, i.e., dogmatically, without grounding or explanation. Of course, how a given collection of objects is specified may change in the course of time, but my analysis of the role logical constants play in producing logical consequences led me to arrive at a criterion of logical constanthood whose 1st-order extension far exceeds the standard selection. More specifically, I showed that if we characterize logical consequence as necessary, formal, topic neutral, indifferent to differences between individuals, etc., then this characterization, restricted to languages of the 1st-level, is not adequately systematized by the standard 1st-order system. A richer system (or family of systems), with new logical constants, is required to fully capture it.
The Bounds of Logic had as its goal a critical, systematic and constructive understanding of logic. As such it aimed at maximum neutrality vis-a-vis epistemic, metaphysical and metamathematical controversies. But a conception of logic does not exist in a vacuum. Eventually our goal is to produce an account of logic that answers the needs of, contributes to the development of, and is supported by, a broader epistemology. In this paper I would like to make a first step in this direction. I will begin with an outline of a model of knowledge whose basic principles are based on the early Quine. I will identify, and offer independent justification for, the special requirements this model sets on an adequate conception of logic. Finally, I will show how, by satisfying these requirements, the conception of logic delineated in The Bounds of Logic (and the related papers) can naturally be incorporated in this epistemic model.
- Fernando Ferreira ‘Amending Frege's Grundgesetze der Arithmetik [draft]’
FregeのLogical Systemを救い出すtechnicalな試み。この手のtechnicalな論文で、よく引用されている重要なものも大概集めてます。Abstractは以下のよう。
Frege's Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, second-order logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative second-order fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
- Thomas Ricketts ‘Logic, Logicism, and the Context Principle in Frege's Grundlagen’
文脈原理は意味の理論の原理ではなく、数概念を分析する際にその内実と表現を与える原理だと解する試み。Introの一部はこう。
he[=Frege] sets forth [the following principles] in the introduction:
- to separate sharply the psychological from the logical, the subjective from the objective;
- to ask after the meaning of a word only in the context of a sentence, not in isolation;
- to keep in mind the distinction between concept and object.
The pivotal principle is the second one, Frege’s famed Context Principle. I hold that it encapsulates Frege’s view of logical segmentation that is a part of his quantificational understanding of logical generality. As such, I shall argue that the Context Principle immediately leads to the type-theoretically stratified view of quantificational generality summarized in the third principle. On this interpretation, the Context Principle is not a principle in theory of meaning that in any way justifies Frege’s introduction of designating proper names for extensions. Rather, the Context Principle gives shape and substance to Frege’s task of analyzing the concept of number. It constrains what will count as an analysis; it leads Frege to logical resources for his attempted analysis; finally, it prompts the introduction of extensions, for only in this way can Frege bring those logical resources to bear on the analytic task at hand.
This paper has four sections. The first section sketches an interpretation of Frege’s three fundamental principles, concentrating on his view of logical segmentation. The second section considers the application of the three principles to Frege’s thesis that statements of number predicate something of concepts. The third section does the same for the thesis that numbers are self-subsistent objects. The fourth section briefly considers Frege’s introduction of extensions.
